Question

Show that the function$\mathit{\delta }$ in Example 6 has property (ii) in the definition of a Euclidean domain in the case when$\mathit{b}\mathbf{<}\mathbf{0}$ . [Hint: Apply the Division Algorithm with a as dividend androle="math" localid="1653649151508" $\left|\mathit{b}\right|$ as divisor. Then modify the result.]

Short answer

It has been proved that$\delta$ holds the mentioned property for$b<0$ .

Step-by-step solution

Given that

A function $\delta$is given by $\delta \left(a\right)=\left|a\right|$

(ii) property of Euclidean Domain is given as

If$a,b\in R$ and$b\ne 0_R$ then there exists$q,r\in R$ such that$a=bq+r$ and either$r=0_R$ or$\delta \left(r\right)<\delta \left(b\right)$ .

Prove that δ has the property

As per the hint apply the division algorithm to a and $\left|b\right|$.

This gives $a=\left|b\right|q+r$with $0\le r<\left|b\right|$.

Now, it is given that $b<0$, thus $a=b\left(-q\right)+r$.

Since $r\ge 0$this implies $r=\left|r\right|=\delta \left(r\right)$and $\delta \left(b\right)=\left|b\right|$

Thus $0\le \delta \left(r\right)<\delta \left(b\right)$.

Hence property holds for$b<0$

Conclusion

Thus, it can be concluded that that$\delta$ holds the Euclidean property for$b<0$ .